General guidelines

• Causal effects are linked to specific populations
• In many epidemiologic studies, you end up analyzing a cohort that’s different from your original cohort
  • Survival analysis: we analyze uncensored individuals (we don’t see the outcome in censored individuals).
  • Case-control studies: we analyze individuals who got the outcome and a sample of patients who did not get the outcome.

General guidelines

• The hope is that the estimate we get in this subset is the same as the one we would’ve had if we did the estimation on the original cohort.
• If this is not the case ➔ Selection bias
• In all the coming DAGs, selection is represented as a variable with a square around it.
• The square here does not mean statistical adjustment. It means that the analysis is done on one stratum of the selection variable.
  • DAGs are not just useful in helping identifying adjustment covariates. They can help in the design as well.

Selection bias under the null

• This type of selection bias can arise regardless of the true treatment effect being null or not.
• Conditioning on a collider (or a descendant of a collider) is necessary for this bias to happen.
• This is the topic of this book and Hernan’s famous paper in 2004 (Hernán, Hernández-Díaz, and Robins 2004)

Selection bias off the null

• This bias is not covered in the chapter.

Example 1

• \(A\) : Treatment

• \(Y\) : Fetal malformation

• \(C\) : Live birth

  We don’t have \(Y\) for dead fetuses, so we essentially restricting our analysis to living fetuses.

Example 1

• Our regression will give us this quantity

\[ \frac{Pr[Y = 1|A = 1, C = 0]}{ Pr[Y = 1|A = 0, C = 0]} \]

• Is this a valid estimate for our estimand?

  \[\frac{Pr[Y^{ a=1} = 1]}{Pr[Y^{ a=0} = 1]} \]

  The answer is no, because we have association transmitted through the path \(A \rightarrow C \leftarrow Y\)

Example 2

• \(A\) : Treatment

• \(Y\) : Fetal malformation

• \(C\) : Live birth

• \(S\): Parental grief

A descendant of a collider is as dangerous as the collider itself.

Example

Example

• Remember, \(C\) is not a variable we put in a regression model. It’s a part of how your analyzed data was formed.

• In this example, \(A\) can show favorable result not because it’s actually effective in reducing mortality, but because it caused sick people to leave the study. Although in reality, \(A\) and \(Y\) are not associated.

• The previous DAG is an example of selection bias due to differential loss-to-follow-up or informative censoring.

Similar DAGs for differential loss-to-follow-up

• These DAGs are modified versions of Figure 8.3

  • For instance, in figure 8.4, the association between \(A\) and \(L\) is represented by mediation while in figure 8.5 presented by a backdoor path \(A \leftarrow W \rightarrow C\) and presented by both in figure 8.6

DAG

• \(E\): Estrogen use

• \(D\): CHD

• \(F\): Hip fracture

• \(C\): Selection into the study

Matched case-control designs are inherently biased

The structural definition of selection bias

Selection bias to refer to all biases that arise from conditioning on a common effect of two variables, one of which is either the treatment or a cause of treatment, and the other is either the outcome or a cause of the outcome.

Selection bias, similar to confounding bias, is a violation of the exchangeability assumption.

Common examples of selection bias

• Differential loss to follow-up or informative censoring.

• Missing data bias, or non-response bias.

• Healthy worker bias.

• Self-selection bias or volunteer bias.

• Selection affected by treatment received before study entry aka prevalent-user bias

• Immortal-time bias is a mix of selection and misclassification bias.

Which designs are prone to selection bias?

• All of them, even randomized experiments.

• Randomization fixes confounding but not selection.

• Selection bias is more likely to occur with designs that are built on selection by default i.e. case-control design

  • Friendly advice, whenever you have the full cohort, please don’t conduct a case-control study.

Which analyses are prone to selection bias?

• Conventional covariate adjustment in treatment-confounder feedback setting.

• Cox regression.

Selection without bias

• RCTs are conducted among volunteers willing to enter the experiment. So those volunteers select into the trial.

• However, this is not what we mean here by selection bias.

• Based on our definition, the selection variable should be a common effect of the treatment or a cause of the treatment and the outcome or cause of the outcome.

• Since volunteering participation happened before treatment assignment, there is no bias.

• The self-selection bias we mentioned earlier is about agreeing to continue in the trial after being treated.

Example

Advantages of using the structural approach

Important reminder

• DAGs ignore the magnitude or direction of selection bias and confounding.
• It is possible that some noncausal paths opened by conditioning on a collider are weak and thus induce little bias.
• It is not an “all or nothing” issue, in practice, it is important to consider the expected direction and magnitude of the bias

In conclusion, we have two issues:

• The estimand changed.

• Selection bias

New estimand

• Similar to the interaction chapter, we will view selection or censoring as an intervention. If we are able to satisfiy the causal identification assumption with \(c\), then this estimand can be estimated using observed data

\[\frac{Pr[Y^{ a=1,c=0} = 1]}{Pr[Y^{ a=0,c=0} = 1]} \]

• This reads as the effect of \(A\) on \(Y\) had everyone got \(A\) and remained uncensored vs everyone not getting \(A\) and remained uncensored.

• Weighting can be a good approach to achieve this (See example).